Learn Icelandic: Hún syndir alltaf best af öllum.

It is an adverb because it describes the verb syndir (swims). The base adverb is vel (well), the comparative is betur (better), and the superlative is best (best). Because it is an adverb and not an adjective, it does not change its ending to match the feminine pronoun hún.

Why is öllum in the dative case?

It is governed by the preposition af. When you want to say best 'of' all or 'out of' all, Icelandic uses af followed by the dative case. Öllum is the dative plural form of the pronoun allur (all).

Could I just say Hún syndir best allra instead of using af?

Yes, you can! Best allra uses the genitive plural of allur and means exactly the same thing: best of all. Both best af öllum and best allra are very common and natural ways to express this in Icelandic.

How would the sentence change if I wanted to say she always swims 'better' than everyone?

You would use the irregular comparative adverb betur (better) along with the word en (than). The sentence would become: Hún syndir alltaf betur en allir. Notice that allir is in the nominative case here because it is the subject of an implied second verb (better than everyone swims).